<h2>Problem 66</h2>
<div style="color:#666;font-size:80%;">26 March 2004</div><br />
<div class="problem_content">
<p>Consider quadratic Diophantine equations of the form:</p>
<p style='text-align:center;'><i>x</i><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> &ndash; D<i>y</i><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 1</p>
<p>For example, when D=13, the minimal solution in <i>x</i> is 649<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> &ndash; 13<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />180<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 1.</p>
<p>It can be assumed that there are no solutions in positive integers when D is square.</p>
<p>By finding minimal solutions in <i>x</i> for D = {2, 3, 5, 6, 7}, we obtain the following:</p>
<p style='margin-left:20px;'>3<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> &ndash; 2<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />2<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 1<br />
2<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> &ndash; 3<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />1<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 1<br />
<span style='color:#dd0000;font-weight:bold;'>9</span><img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> &ndash; 5<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />4<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 1<br />
5<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> &ndash; 6<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />2<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 1<br />
8<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> &ndash; 7<img src='images/symbol_times.gif' width='9' height='9' alt='&times;' border='0' style='vertical-align:middle;' />3<img src="" style="display:none;" alt="^(" /><sup>2</sup><img src="" style="display:none;" alt=")" /> = 1</p>
<p>Hence, by considering minimal solutions in <i>x</i> for D <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 7, the largest <i>x</i> is obtained when D=5.</p>
<p>Find the value of D <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> 1000 in minimal solutions of <i>x</i> for which the largest value of <i>x</i> is obtained.</p>

</div><br />
